A domestic manufacturer of watches purchases quart crystals...

A domestic manufacturer of watches purchases quart crystals from a Swiss firm. The crystals are shipped in lots of $1000 .$ The acceptance sampling procedure uses 20 randomly selected crystals. $\begin{array}{l}{\text { a. Construct operating characteristic curves for acceptance numbers of } 0,1, \text { and } 2 .} \\ {\text { b. } \text { If } p_{0} \text { is } .01 \text { and } p_{1}=.08 \text { , what are the producer's and consumer's risks for each sam- }} \\ {\text { pling plan in part (a)? }}\end{array}$

A domestic manufacturer of watches purchases quart crystals from a Swiss firm. The
crystals are shipped in lots of $1000 .$ The acceptance sampling procedure uses 20 randomly
selected crystals.
$\begin{array}{l}{\text { a. Construct operating characteristic curves for acceptance numbers of } 0,1, \text { and } 2 .} \\ {\text { b. } \text { If } p_{0} \text { is } .01 \text { and } p_{1}=.08 \text { , what are the producer's and consumer's risks for each sam- }} \\ {\text { pling plan in part (a)? }}\end{array}$

Key Concepts

Sampling Plan Design

Sampling plan design involves determining the sample size and the acceptance number (the maximum number of defective items allowed in the sample) for a lot. The design of the sampling plan is based on balancing producer's risk and consumer's risk, and it often uses statistical distributions (like the binomial model) to predict the probabilities of different outcomes.

Consumer's Risk

Consumer's risk, also known as Type II error, is the probability of accepting a lot that does not meet the quality standards. It reflects the likelihood that a consumer receives a defective lot, highlighting the risk associated with the sampling plan from the consumer’s perspective.

Operating Characteristic Curve

An operating characteristic (OC) curve plots the probability of accepting a lot against various levels of the fraction defective. This curve helps in evaluating the performance of a sampling plan by illustrating how likely it is to accept lots of different quality levels, thus providing insight into the risks accepted by both producers and consumers.

Acceptance Sampling

Acceptance sampling is a statistical quality control procedure used to decide whether to accept or reject a lot based on a sample drawn from it. The method is particularly useful when testing every item in a lot is impractical, and it balances the cost of inspection with the need to maintain quality standards.

Producer's Risk

Producer's risk, also known as Type I error, is the probability of rejecting a lot that actually meets the quality standards. This risk represents the potential loss or cost to the manufacturer when a good lot is erroneously deemed unacceptable based on the sampling plan.

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